It's a bit of a circular question & answer. A device that needs a regulated power supply is...well...any device that needs regulated power.
For one possible example, take a personal computer. There are components within which need very tighty-controlled voltages. If, say, the power supply hich is supposed to output a steady 3.3V instead varied wildly from 2V to 5V because of fluctuations on the input side of the power supply, then the computer may not operate correctly or might even be damaged.
One of the "nice" things about some types of regulated power supplies is that under certain failure conditions they will self-desctruct rather than output voltages which might damage the device. The upside is the device is saved. The down side is that the supply will often suffer so much damage that it has to be replaced. But it's generally cheaper to buy a new supply than a whole new device.